Gaussian process regression for Bayesian inverse problems

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Mathematical models based on partial differential equations appear everywhere in science and engineering. Parameters in the models, appearing for example in boundary conditions and coefficients, are typically not fully known and have to be estimated from observed data. Accurate reconstruction of the parameters, as well as an estimate of the uncertainty in the reconstruction, are crucial for reliable predictions and risk assessments. An example application is underground carbon storage, where the precise make-up of the environment, such as the location and hydraulic conductivity of different layers of rock, is not fully known and has to be estimated from measurements. For safety reasons, it is crucial to quantify the risk of leaked particles re-entering the human environment.Mathematically, the process of learning unknown model parameters from data, also known as model calibration, can be formulated as the inverse problem to recover unknown parameters from noisy, indirect observations. Following the Bayesian approach, we obtain a posterior distribution for the unknown parameters conditioned on the observed data. Because of the complexity of models involved in modern applications, there is a pressing need to develop efficient algorithms for exploring the posterior.The overall aim of this proposal is to design, analyse and implement computational methods based on Gaussian process regression to solve inverse problems in partial differential equations accurately and efficiently. We will bring together ideas from numerical analysis, approximation theory, statistics and optimisation, to develop sophisticated algorithms that can be applied and adapted across a range of sectors, including huge potential for applications in health (medical imaging) and environment (subsurface modelling, underground carbon storage).